This invention relates to a building air-conditioning device that uses treated air that is admitted in a volume that is created between walls, a ceiling and a sealed false ceiling, but that comprises a thin space that is reserved over all or part of its periphery. This treated air, after having cooled or heated the false ceiling, which in its turn radiates to the building, diffuses in the space between the wall and the false ceiling to ensure the remainder of the heating or cooling of the building, as well as the mixing and the renewal of the air of the building.